Family Fibration of a Fibration

Let $\mathcal{E}\to\mathcal{B}$ be a Cartesian Fibration. The family fibration $\mathrm{Fam}(\mathcal{E})\to\mathcal{B}$ is defined as the Pullback of $\mathcal{E}$ along the functor $\mathrm{Dom}:\mathcal{B}^{\to}\to\mathcal{B}$, Composed with the Codomain Fibration.

We can also give an explicit description of $\mathrm{Fam}(\mathcal{E})$:

• An object over $I:\mathcal{B}$ is a pair of a slice $(X,f:X\to I)$ and an object $P:\mathcal{E}_{X}$.
• A morphism over $u:\mathcal{B}(I,J)$ between $(X,f,P)$ and $(Y,g,Q)$ is a morphism $u^{\sharp}:X\to Y$ such that $u\circ f=g\circ u^{\sharp}$, along with a map $\mathcal{E}_{u^{\sharp}}(P,Q)$.
• Identities and composites are given in the obvious way.

As usual, we want to think of the Arrow Category as a sort of "category of families" internal to to $\mathcal{B}$; this becomes clear when we look at the special case where $\mathcal{B}$ is the Category of Sets, where the Codomain Fibration is the Family Fibration $\mathrm{Fam}(\mathrm{Sets})\to\mathrm{Sets}$. This means that we should think about this fibration as a family internal to $\mathcal{B}$, and an element of $\mathcal{E}$ over the "total space" of the family. If we specialize this again to $\mathrm{Sets}$, this means that we have a family $X:I\to\mathcal{U}$, and an element of $\mathcal{E}_{\Sigma I.X}$, so this is a way of placing a fibration over a bundle in $\mathcal{B}$.

As a Monad

This construction induces a Monad on $\mathrm{Fib}(\mathcal{B})$; the unit $\eta:\mathcal{E}\to\mathrm{Fam}(\mathcal{E})$ sends an object $P:\mathcal{E}_{X}$ to the slice $(X,\mathrm{id},P)$, and the join sends $(X,u,(Y,v,P))$ to $(X,uv,P)$, modulo some reindexing.

References

• Fibered Categories a La Jean Benabou, 6.2; note that this uses the somewhat annoying notation of $P_{\mathcal{B}}$ to denote the Codomain Fibration.